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%I #10 Dec 09 2014 00:40:29
%S 1,6,107,3508,171741,11280842,933014767,93212094024,10925496633401,
%T 1470493880790382,223555405538724819,37892802280129883324,
%U 7086076189702624109653,1449303152891376476830962,321848482510755456019058519,77124029495405859198280522768
%N E.g.f.: exp(7*x*G(x)^6) / G(x) where G(x) = 1 + x*G(x)^7 is the g.f. of A002296.
%F Let G(x) = 1 + x*G(x)^7 be the g.f. of A002296, then the e.g.f. A(x) of this sequence satisfies:
%F (1) A'(x)/A(x) = G(x)^6 + 5*G'(x)/G(x).
%F (2) A(x) = F(x/A(x)^6) where F(x) is the e.g.f. of A251697.
%F (3) A(x) = Sum_{n>=0} A251697(n)*(x/A(x)^6)^n/n! where A251697(n) = (5*n+1) * (6*n+1)^(n-2) * 7^n .
%F (4) [x^n/n!] A(x)^(6*n+1) = (5*n+1) * (6*n+1)^(n-1) * 7^n .
%F a(n) = Sum_{k=0..n} 7^k * n!/k! * binomial(7*n-k-2,n-k) * (6*k-1)/(6*n-1) for n>=0.
%F Recurrence: 72*(2*n-1)*(3*n-2)*(3*n-1)*(6*n-5)*(6*n-1)*(588245*n^6 - 6117748*n^5 + 26651100*n^4 - 62321728*n^3 + 82554122*n^2 - 58646294*n + 17291583)*a(n) = 7*(69206436005*n^12 - 996572678472*n^11 + 6516703994430*n^10 - 25624338676965*n^9 + 67604945463195*n^8 - 126360374558838*n^7 + 171960790012102*n^6 - 171911061779835*n^5 + 125050872537045*n^4 - 63802357502870*n^3 + 20814954345360*n^2 - 3329274812661*n - 3763584000)*a(n-1) - 823543*(588245*n^6 - 2588278*n^5 + 4886035*n^4 - 5129908*n^3 + 3141733*n^2 - 958104*n - 720)*a(n-2). - _Vaclav Kotesovec_, Dec 07 2014
%F a(n) ~ 5 * 7^(7*n-3/2) / 6^(6*n-1/2) * n^(n-1) / exp(n-1). - _Vaclav Kotesovec_, Dec 07 2014
%e E.g.f.: A(x) = 1 + 6*x + 107*x^2/2! + 3508*x^3/3! + 171741*x^4/4! + 11280842*x^5/5! +...
%e such that A(x) = exp(7*x*G(x)^6) / G(x)
%e where G(x) = 1 + x*G(x)^7 is the g.f. of A002296:
%e G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 +...
%t Table[Sum[7^k * n!/k! * Binomial[7*n-k-2,n-k] * (6*k-1)/(6*n-1),{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Dec 07 2014 *)
%o (PARI) {a(n)=local(G=1); for(i=0, n, G=1+x*G^7 +x*O(x^n)); n!*polcoeff(exp(7*x*G^6)/G, n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n) = sum(k=0, n, 7^k * n!/k! * binomial(7*n-k-2,n-k) * (6*k-1)/(6*n-1) )}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A251577, A251697, A002295.
%Y Cf. Variants: A243953, A251663, A251664, A251665, A251666, A251668, A251669, A251670.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 07 2014
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